= Solution
Set
$$
U=e^{-X},
\qquad
V=e^{-Y}.
$$
These are bounded random variables with values in $(0,1]$, and for every positive integer $n$,
$$
\mathbb E[U^n]=\psi_X(n)=\psi_Y(n)=\mathbb E[V^n].
$$
Part c shows that $U$ and $V$ have the same law. Since $x\mapsto e^{-x}$ is injective on $[0,\infty)$ with inverse $u\mapsto-\log u$, taking the <pushforward measure> by this inverse shows that $X$ and $Y$ have the same law. This proves the <uniqueness theorem for Laplace transforms of nonnegative random variables>.
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